# Аналитическое дифференцирование функций

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131``` ```(define op car) (define args cdr) (define first-arg cadr) (define second-arg caddr) (define other-args cddr) (define (bisection func array) (cons (filter func array) (filter (lambda (el) (not (func el))) array))) (define (has-0 L) (cond ((null? L) #f) ((and (number? (car L)) (= (car L) 0)) #t) (else (has-0 (cdr L))))) (define (constant? exp var) (and (not (pair? exp)) (not (eq? exp var)))) (define (same-var? exp var) (and (not (pair? exp)) (eq? exp var))) (define (sum? exp) (and (pair? exp) (eq? (op exp) '+))) (define (make-sum . a) (let ((a (filter (lambda (n) (not (and (number? n) (= n 0)))) a))) (cond ((null? (cdr a)) (car a)) (else (let ((numbers (bisection number? a))) (cond ((= 0 (length (cdr numbers))) (apply + (car numbers))) (else (let ((sums (bisection sum? a))) (cons '+ (append (reduce append '() (map args (car sums))) (cdr sums))))))))))) (define (diff? exp) (and (pair? exp) (eq? (op exp) '-))) (define (make-diff a1 a2) (cond ((and (number? a1) (number? a2)) (- a1 a2)) ((and (number? a1) (= a1 0)) (make-product -1 a2)) ((and (number? a2) (= a2 0)) a1) (else (list '- a1 a2)))) (define (product? exp) (and (pair? exp) (eq? (op exp) '*))) (define (make-product . m) (let ((m (filter (lambda (n) (not (and (number? n) (= n 1)))) m))) (cond ((null? m) 1) ((null? (cdr m)) (car m)) ((has-0 m) 0) (else (let ((numbers (bisection number? m))) (cond ((= 0 (length (cdr numbers))) (apply * (car numbers))) (else (let ((products (bisection product? m))) (cons '* (append (reduce append '() (map args (car products))) (cdr products))))))))))) (define (division? exp) (and (pair? exp) (eq? (car exp) '/))) (define (make-division d1 d2) (cond ((and (number? d2) (= d2 1)) d1) ((and (number? d2) (= d1 0)) 0) (else (list '/ d1 d2)))) (define (func? exp) (not (or (not (pair? exp)) (eq? (op exp) '+) (eq? (op exp) '-) (eq? (op exp) '*) (eq? (op exp) '/)))) (define (make-func f1 f2 var) (make-product (derive f2 var) (cond ((eq? f1 'sin) (list 'cos f2)) ((eq? f1 'cos) (make-product -1 (list 'sin f2))) ((eq? f1 'ln) (make-division 1 f2)) ((eq? f1 'sqr) (make-product 2 f2))))) (define (derive exp var) (cond ((constant? exp var) 0) ((same-var? exp var) 1) ((sum? exp) (apply make-sum (map (lambda (exp) (derive exp var)) (args exp)))) ((diff? exp) (make-diff (derive (first-arg exp) var) (derive (second-arg exp) var))) ((product? exp) (cond ((not (pair? (cddr exp))) (make-sum (make-product (first-arg exp) (derive (second-arg exp) var)) (make-product (second-arg exp) (derive (first-arg exp) var)))) (else (make-sum (make-product (first-arg exp) (derive (apply make-product (other-args exp)) var)) (make-product (apply make-product (other-args exp)) (derive (first-arg exp) var)))))) ((division? exp) (make-division (make-diff (make-product (derive (first-arg exp) var) (second-arg exp)) (make-product (derive (second-arg exp) var) (first-arg exp))) (make-func 'sqr (make-product (first-arg exp) (second-arg exp)) var))) ((func? exp) (make-func (op exp) (first-arg exp) var)))) ; testing (define foo '(sin (ln (sqr x)))) (define bar '(+ x x x x x)) (define baz '(/ (* x x x) (ln x))) (derive baz 'x) ```